Related papers: Shadow-complexity and trisection genus

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a…

The Heegaard genus of a 3-manifold, as well as the growth of Heegaard genus in its finite sheeted cover spaces, has extensively been studied in terms of algebraic, geometric and topological properties of the 3-manifold. This note shows that…

The paper contains a talk given by the author at the Banach Center in Spring 1995. It recapitulates author's approach to construction of topological invariants of the Reshetikhin-Turaev-Witten type of 3- and 4-dimensional manifolds in the…

The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in (R. Vigara, Representaci\'on de 3-variedades por esferas de Dehn rellenantes, PhD Thesis, UNED 2006). They are topological invariants that give a…

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to…

We introduce a natural-valued complexity c(X) for pairs X=(M,L), where M is a closed orientable 3-manifold and L is a link contained in M. The definition employs simple spines, but for well-behaved X's we show that c(X) equals the minimal…

We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show…

Topological complexity is a numerical homotopy invariant that measures the instability of motion planning in a space. To study the topological complexity of non-simply connected spaces, Costa and Farber introduced a cohomology class whose…

Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time…

This survey covers some of the results contained in the papers by Costantino, Geer and Patureau (https://arxiv.org/abs/1202.3553) and by Blanchet, Costantino, Geer and Patureau (https://arxiv.org/abs/1404.7289). In the first one the authors…

The triangulation complexity of a compact 3-manifold is the minimal number of tetrahedra in any triangulation of the 3-manifold. We compute the triangulation complexity of all elliptic 3-manifolds and all sol 3-manifolds, to within a…

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of…

We show that the problem of deciding whether a knot in a fixed closed orientable 3-dimensional manifold bounds a surface of genus at most $g$ is in co-NP. This answers a question of Agol, Hass, and Thurston in 2002. Previously, this was…

We construct an infinite family $\{ C_{n,k}\}_{k=1}^{\infty}$ of corks of Mazur type satisfying $2n\leq \mathrm{sc}^{\mathrm{sp}}(C_{n,k})\leq O(n^{3/2})$ for any positive integer $n$. Furthermore, using these corks, we construct an…

We present upper and lower bounds for symmetrized topological complexity $TC^\Sigma(X)$ in the sense of Basabe-Gonz\'alez-Rudyak-Tamaki. The upper bound comes from equivariant obstruction theory, and the lower bounds from the cohomology of…

The notions Golodness and tightness for simplicial complexes come from algebra and geometry, respectively. We prove these two notions are equivalent for 3-manifold triangulations, through a topological characterization of a polyhedral…

We prove a minimum degree version of the Kruskal--Katona theorem: given $d\ge 1/4$ and a triple system $F$ on $n$ vertices with minimum degree at least $d\binom n2$, we obtain asymptotically tight lower bounds for the size of its shadow.…

Virtual $3$-manifolds were introduced by S.V. Matveev in 2009 as natural generalizations of the classical $3$-manifolds. In this paper, we introduce a notion of complexity of a virtual $3$-manifold. We investigate the values of the…

Every element in the first cohomology group of a 3--manifold is dual to embedded surfaces. The Thurston norm measures the minimal `complexity' of such surfaces. For instance the Thurston norm of a knot complement determines the genus of the…

Let $G$ be a finite group, and let $X$ be a smooth, orientable, connected, closed 4-dimensional $G$-manifold. Let $\mathcal{S}$ be a smooth, embedded, $G$-invariant surface in $X$. We introduce the concept of a $G$-equivariant trisection of…